Let $A$ be an $n \times n$ invertible complex matrix. Let $Gr(k)=Gr(k,\mathbb{C}^n)$ be the complex $k$-Grassmannian, $1\leq k \leq n$. Since $A$ is invertible, it maps a $k$-dimensional subspace to a $k$-dimensional subspace, so it gives a function (which I'll call $A_k$) on $Gr(k)$. The fact that matrices have eigenvalues lets us deduce that $A$ has a $k$-dimensional invariant subspace - i.e. this map $A_k$ has a fixed point.

My question is this: is there a fixed point theorem on $Gr(k)$ (which ideally does not in any way depend on the existence of eigenvalues) that we can invoke to arrive at the same conclusion?